Intervals as Semitones
It can be helpful for students to see intervals both as scale degree relationships, and also as a specific number of half steps or semitones. I've heard many who say semitone counts are difficult to memorize. Here's a formula that makes it easy!
How to remember semitone counts:
There are 12 semitones in a one-octave chromatic scale. There are 8 notes in a one-octave major scale, where 4 intervals are perfect, and 4 have major/minor qualities. So, think of there being 4 additional semitones in a chromatic scale, versus notes in an octave. 8+4=12. 12 semitones total.
A tritone splits the semitone octave in half. So, there are 6 semitones in a tritone (12÷2=6).
A tritone is an augmented 4th and a diminished 5th. So, a perfect 4th is 5 semi-tones (one less than a tritone), and a perfect 5th is 7 semitones (one more than a tri-tone).
A half step and whole step are 1 semitone and 2 semitones, respectively. So, a minor 2nd is 1 semitone and a major 2nd is 2 semitones.
Major and minor 7ths are reversed from 2nds. A half step below 12 is 11 and whole step below 12 is 10. So, a major 7th is 11 semitones and a minor 7th is 10 semitones.
That leaves 4 intervals remaining – the only 4 you could memorize: major and minor 3rd, and major and minor 6th. 3 4; 8 9. A major 3rd is 4 semi-tones and minor 3rd is 3 semitones. A major 6th is 9 semitones and minor 6th is 8 semi-tones.
But... if you remember that a major 3rd is one less than a perfect 4th, or 4 semitones, then a minor 3rd is one less than a major 3rd, or 3 semitones. Likewise, a minor 6th is one more than perfect 5th, or 8 semitones, and a major 6th is one more than a minor 6th, or 9 semitones. So, you really don’t have to memorize anything!